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Mirrors > Home > ILE Home > Th. List > cnvi | GIF version |
Description: The converse of the identity relation. Theorem 3.7(ii) of [Monk1] p. 36. (Contributed by NM, 26-Apr-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
Ref | Expression |
---|---|
cnvi | ⊢ ◡ I = I |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 2560 | . . . . 5 ⊢ 𝑥 ∈ V | |
2 | 1 | ideq 4488 | . . . 4 ⊢ (𝑦 I 𝑥 ↔ 𝑦 = 𝑥) |
3 | equcom 1593 | . . . 4 ⊢ (𝑦 = 𝑥 ↔ 𝑥 = 𝑦) | |
4 | 2, 3 | bitri 173 | . . 3 ⊢ (𝑦 I 𝑥 ↔ 𝑥 = 𝑦) |
5 | 4 | opabbii 3824 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ 𝑦 I 𝑥} = {〈𝑥, 𝑦〉 ∣ 𝑥 = 𝑦} |
6 | df-cnv 4353 | . 2 ⊢ ◡ I = {〈𝑥, 𝑦〉 ∣ 𝑦 I 𝑥} | |
7 | df-id 4030 | . 2 ⊢ I = {〈𝑥, 𝑦〉 ∣ 𝑥 = 𝑦} | |
8 | 5, 6, 7 | 3eqtr4i 2070 | 1 ⊢ ◡ I = I |
Colors of variables: wff set class |
Syntax hints: = wceq 1243 class class class wbr 3764 {copab 3817 I cid 4025 ◡ccnv 4344 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-br 3765 df-opab 3819 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 |
This theorem is referenced by: coi2 4837 funi 4932 cnvresid 4973 fcoi1 5070 ssdomg 6258 |
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